Generalized Quadrangles with a Spread of Symmetry

We present a common construction for some known infinite classes of generalized quadrangles. Whether this construction yields other (unknown) generalized quadrangles is an open problem. The class of generalized quadrangles obtained this way is characterized in two different ways. On the one hand, they are exactly the generalized quadrangles having a spread of symmetry. On the other hand, they can be characterized in terms of the group of projectivities with respect to a spread. We explore some properties of these generalized quadrangles. All these results can be applied to the theory of the glued near hexagons, a class of near hexagons introduced by the author in De Bruyn (1998) On near hexagons and spreads of generalized quadrangles, preprint.